Let $K$ be a characteristic zero algebraic function field with a valuation $\nu$. Let $L$ be a finite extension of $K$ and $\omega$ be an extension of $\nu$ to $L$. We establish that the valuation ring $V_{\omega }$ of $\omega$ is essentially finitely generated over the valuation ring $V_{\nu }$ of $\nu$ if and only if the initial index $𝜖(\omega |\nu )$ is equal to the ramification index $e(\omega |\nu )$ of the extension. This gives a positive answer, for characteristic zero algebraic function fields, to a question posed by Hagen Knaf.

中文翻译：

---

特征零代数函数域中估值环的本质有限生成

让$ķ$是具有估值的特征零代数函数场$\nu$. 让$\mathrm{大号}$是的有限扩展 $ķ$和$\omega$成为的延伸$\nu$到$\mathrm{大号}$. 我们建立了估值环${五}_{\omega }$的$\omega$本质上是在估值环上有限生成的${五}_{\nu }$的$\nu$当且仅当初始索引$𝜖(\omega |\nu )$等于分枝指数$e(\omega |\nu )$的扩展。对于 Hagen Knaf 提出的问题，对于特征零代数函数域，这给出了肯定的答案。